1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Finding a cubic polynomial that attains a max/min value over an open interval

  1. Feb 5, 2013 #1
    1. The problem statement, all variables and given/known data

    Give an example of a cubic polynomial, defined on the open interval (-1,4), which reaches both its maximum and minimum values.

    2. Relevant equations


    3. The attempt at a solution

    I can see that I would need a function such that there is some f(a) and f(b) in (f(-1),f(4)) such that f(a) >= all f(x) for x in (-1,4) and f(b) <= all f(x) for x in (-1,4). I used an online tool to adjust the coefficients of a cubic until I got what I needed. But I have no idea how to do this by myself. All that I can think of is to somehow use the fact that at extrema, a function's derivative is zero.
  2. jcsd
  3. Feb 5, 2013 #2


    Staff: Mentor

    Write your cubic as y = (x - a)(x - b)(x - c). The x-intercepts are at (a, 0), (b, 0), and (c, 0). Without too much effort you can put in values for a, b, and c so that all three intercepts are in the interval (-1, 4), with a local maximum between a and b, and a local minimum between b and c.
  4. Feb 5, 2013 #3

    Ray Vickson

    User Avatar
    Science Advisor
    Homework Helper

    You could find just any old cubic p(x) that has distinct maxima and minima, then shift and re-scale x until the max and min lie in your interval.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Finding a cubic polynomial that attains a max/min value over an open interval